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In an earlier paper (B. J.
Gardner, Pacific J. Math., 33 (1970), 109-116) the torsion classes of abelian groups
which are closed under pure subgroups were characterized, and §§3-6 of the
present paper are devoted to generalizations of results appearing there. If
𝒞 is a homomorphically closed class of objects in an abelian category, a
subobject A of an object B is called 𝒞-pure if it is a direct summand of every
intermediate subobject X for which X∕A ∈𝒞. (This terminology is due to C. P.
Walker). In particular, 𝒞 may be a torsion class. The following question is
investigated: If 𝒯− and 𝒰 are torsion classes of abelian groups, when is 𝒯− closed
under 𝒰-pure subgroups? Although ordinary purity is not 𝒰-purity for any
torsion class 𝒰, a torsion class 𝒯 is closed under pure subgroups if and only if
it is closed under 𝒯0-pure subgroups, where 𝒯0 is the class of all torsion
groups.
In §5, for an arbitrary torsion theory (𝒰,𝒢) a rank function ( 𝒰-rank) is defined
for nonzero groups in 𝒢. It is: shown that every torsion class closed under 𝒰-pure
subgroups. is determined by its intersection with 𝒰 and the groups of 𝒰-rank 1 it
contains. When 𝒰 = 𝒯0, the groups with 𝒰-rank 1 are the rational groups, so the
earlier results for ordinary purity suggest that in general some refinement of the
representation should be possible.
A further special case of the general problem is als α solved: Let X and Y be
rational groups, T(X),T(Y ) the smallest torsion classes containing them. If X is a
subring of the rationals then T(X) is always closed under T(Y )-pure subgroups;
if not, the condition is satisfied if and only if X has a greater type than
Y.
§7 is devoted to proving the following result: A torsion class is closed under
countable direct products, i.e. direct products of countable sets of groups, if and only
if it is determined by torsion-free groups.
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